Denmark is divided into three geographical parts. Jutland is the part that is attached to continental Europe and which shares a border with Germany. Funen is the large island to the east of Jutland. Zealand is the even larger island to the east again after Funen, and is where the capital city of Copenhagen is situated. Knowing this will help you understand more quickly where different destinations are located.
Denmark has not adopted the euro and has retained its own currency, the Danish Crown. At the time of writing, you can expect to find that things in Denmark cost up to 30% more than in other European countries using the euro.
Danes in general have high standards concerning civic responsibility and take it for granted that others do too. So for example, there are no ticket barriers in railway stations and it may be unlikely that anybody checks your ticket, because the assumption is that you too have acted responsibly and bought the ticket you need.
Many Danes speak English (possibly German in the South of Denmark) and you may never need to speak a word of Danish. However, just knowing something as simple as “Tak” (meaning thank you) can go a long way to ease communications. Danes are also often quite direct in the way they speak and the way they establish eye contact, while still remaining pleasant and friendly.
The Danish climate can be highly variable. In the summer for example, the weather can be sunny and fine, or wet and distinctly cooler. Danes know this and this is why there is a considerable range of things to do inside (whether in terms of culture, entertainment or sport) as well as outside, just in case.